KC-space

From Topospaces
Jump to: navigation, search
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is a variation of Hausdorffness
View other variations of Hausdorffness | read a survey article on varying Hausdorffness

Definition

Symbol-free definition

A topological space is termed a KC-space if every compact subset of it is closed (here, by compact subset, we mean a subset which is a compact space under the subspace topology).

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Hausdorff space any two distinct points can be separated by disjoint open subsets Hausdorff implies KC KC not implies Hausdorff Weakly Hausdorff space|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
weakly Hausdorff space any subset of it arising as the continuous image of a compact Hausdorff space is closed in it |
US-space KC implies US US not implies KC |
T1 space points are closed KC implies T1 T1 not implies KC |
Kolmogorov space any two points can be distinguished (via T1) (via T1) |